I remember some old movie that was on TV ~30 years ago. A terrorist group broke into some computer room to destroy the data. They shot the monitors to smithereens and ran away.
Considering our IT department replaces computers without moving over our files (like come on, just swap the drives!), I honestly wouldn't be surprised if that's how they'd treat it.
Honestly most unsavvy people don't even realize they can turn their monitors off. Especially if the buttons are behind or under the screen, they wouldn't even know the buttons were there.
There's some older ones where there are actual buttons on the bottom of the screen. Beats me how the people who press them to turn it off manage to press the power button for the PC to turn it on.